Direct products of modules and the pure semisimplicity conjecture. Part II

TL;DR

This paper proves that certain Noether and affine noetherian PI algebras satisfy a product property related to module decompositions, supporting the pure semisimplicity conjecture linking pure global dimension to finite representation type.

Contribution

It establishes that module categories of these algebras have the product property, providing evidence for the pure semisimplicity conjecture in this context.

Findings

Direct product of indecomposable modules decomposes into finitely generated objects only with repeated types.

Rings satisfying the property support the pure semisimplicity conjecture.

Supports the link between pure global dimension and finite representation type.

Abstract

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in \Bbb N} M_n$ of finitely generated indecomposable modules $M_n$ is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the $M_n$. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.